Sample Paper for MA / M.Sc Mathematics of Complex Analysis !!
Sample Paper for MA/M.Sc Mathematics of Complex Analysis
TERM END EXAMINATIONS
M. A. /M. Sc. – MATHEMATICS, YEAR – 1
Duration -3 Hours Max Marks: 60
Note: 1. Attempt any FIVE questions.
2. All questions carry equal marks.
1. (a) Show that function : f(z) = (z ¹ 0), f(0) = 0 is not analytic at z = 0, although C.R. equations satisfied at that point.
(b) Derive the formula for the radius of convergence of the power series an zn f find the same for the series.
2. (a Using Cauchy’s integral formula, prove that where C is the circle |z| = 3
(b) State & prove weiestrass theorem concerning the behaviour of an analytic function near an isolatial essential singularity.
3. (a) State & prove Argument principle
(b) Prove that every polynomial of degree n has no zeros & determine the number of roots of the equation
z8 – 4z5 + z2 – 1 = 0 that lies inside the circle |z| = 1
4. (a) State & prove minimum modulus principle.
(b) Show that cot z =
5. (a) State & prove Taylor’s Theorem.
(b) Expand f(z) = in Lament’s series valid for the regions
| < |z+1| < 2
6. (a) State & prove Fundamental theorem of Algebra with the help of Morera theorem.
(b) Expand log |1+z| is Taylor’s series about z = 0 & determine the region of convergence for the resulting series.
7. (a) There cannot be two different direct analytic continuations of a function
(b) If f(z) is memorphic inside a closed contour c & has no zero on c then
N is the number of zeros & P the number of poles inside C.
8. (a) State & prove Rouche’s theorem,
(b) State & prove Mittag-Leffler Theorem.